The rank of the matrix $\left[\begin{array}{ccc}1 & 0 & 2 \\ 0 & 1 & -2 \\ 1 & -1 & 4 \\ 2 & 2 & 8\end{array}\right]$ is

Let $f$ be a twice differentiable function defined on $R$ such that $f(0)=1$,$f^{\prime}(0)=2$ and $f^{\prime}(x) \neq 0$ for all $x \in R$. If $\left|\begin{array}{ll}f(x) & f^{\prime}(x) \\ f^{\prime}(x) & f^{\prime \prime}(x)\end{array}\right|=0$ for all $x \in R$,then the value of $f(1)$ lies in the interval:

Which of the following statements is false?
$1$. If $A$ is a skew-symmetric matrix of order $5 \times 5$,then the rank of $A$ is less than $5$.
$2$. If $P$ is a non-zero column matrix and $Q$ is a non-zero row matrix,then the rank of $PQ$ is $1$.
$3$. The rank of $\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 7 \end{bmatrix}$ is $2$.
$4$. If the lines $a_r x + b_r y + c_r = 0$ $(r = 1, 2, 3)$ are distinct and intersect at a point,then the rank of $\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}$ is $3$.

$A$ is a singular matrix of order $5$. $B$ is another matrix having the rank $\rho(B)$ equal to the rank $\rho(A)$, and $B$ has a non-zero minor of order $3$. Then which one of the following is true?

If $f: N \to Z$ is defined by $f(n) = \det \begin{vmatrix} n & -1 & -5 \\ -2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3(2k+1) & 3(k+2)+1 \end{vmatrix}$, where $k \in N$ and $\sum_{n=1}^k f(n) = 98$, then $k$ is equal to:

The rank of the matrix $\left[ {\begin{array}{*{20}{c}}4&1&0&0\\3&0&1&0\\6&0&2&0\end{array}} \right]$ is: